Question

Triangle Inequality Theorem Practice
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Question
The measure of two sides of a triangle are 7 and 11. Determine the range of values for the third side of the triangle.(1 point)
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Answer 1

To determine the range of values for the third side of a triangle when the other two sides are given, we can use the Triangle Inequality Theorem. According to this theorem, the sum of the lengths of any two sides of a triangle must be greater than the length of the third side.

Let's denote the sides of the triangle as follows:
- Side 1 = 7
- Side 2 = 11
- Side 3 = n (the unknown side)

According to the Triangle Inequality Theorem, the following three inequalities must be true:

1. \( 7 + 11 > n \)
2. \( 7 + n > 11 \)
3. \( 11 + n > 7 \)

Now, we will solve these inequalities step by step:

1. From the first inequality \( 7 + 11 > n \):
\[
18 > n \quad \text{or} \quad n < 18
\]

2. From the second inequality \( 7 + n > 11 \):
\[
n > 11 - 7 \quad \text{or} \quad n > 4
\]

3. From the third inequality \( 11 + n > 7 \):
\[
n > 7 - 11 \quad \text{or} \quad n > -4
\]
(This condition is always satisfied, since \( n > 4 \) is more restrictive.)

Combining the results of the first two inequalities, we have:
\[
4 < n < 18
\]

So the range of values for the third side \( n \) is:
\[
4 < n < 18
\]